Outline

• Motivating Example
• Working Example
• Model & Visualizations

Motivating Example

Childhood vaccination - major public health concern

• Vaccination animation

Questions

Interactions in animation aren’t realistic - people don’t behave like bacteria in petri dishes!

• How do community beliefs affect vaccination rate?
• How do those beliefs spread in a population?
• How does community structure affect spread of disease from outside source?
• What other factors lead to unvaccinated children?
• Can communities be identified and targeted for immunization programs/initiatives?

Can we use network modeling to answer these questions?

Working Example

Another public health concern - underage drinking & smoking (Steglich, Snijders & Pearson 2010)

Data from “Teenage Friends and Lifestyle Study” (Pearson & Mitchell (2000); Pearson & West (2003))

• Glasgow, Scotland, UK, Feb 1995 - Jan 1997
• Panel data - three waves - aged 12-13 to 14-15
• Each student lists up to 6 school friends
• Smoking & drinking behavior, other lifestyle variables collected
• 160 students in total, only 50 in example data
• We use an even smaller subset of 16 female students

Network Visualization

Model

• Want: to determine how friendships change risky behavior and vice versa over time
• Solution: stochastic actor-oriented models (SAOMs) for networks (Snijders 1996)
  • Stochastic because they model network change over time
  • Actor-oriented because actor-level covariates are incorporated into model parameters i.e. drinking behavior taken into account when modeling network change
  • Network structure can also be taken into account when modeling behavior change

How do SAOMs work?

Just consider modeling network change for now

• Change doesn’t happen all at once, but observations are discrete.
  • Many unobserved changes between discrete time points
  • Model unobserved data as Continuous-time Markov chains (CTMCs)
• Ties are dependent, with a complex dependency structure
  • Condition on wave 1 (need at least 2 observations in data)
  • For steps in CTMC, condition on all current ties

Markov Chain process

• Choose one actor (call it \(i\)) to (potentially) make a change.
  • Probability of actor choice depends on rate function
• Actor \(i\) tries to optimize some utility function by changing ties or not
  • Choose a “best” friend to have or a “worst” friend to cut loose

Rate Function

Rate at which actors are selected to act

• Simple rate parameter, \(\alpha\), for all actors
  • Can be a function of network statistics or actor covariates, but keep it simple for our purposes
• Waiting time for one actor to act \(\sim Exp\left(\frac{1}{\alpha}\right)\)
• Waiting time for any actor to act \(\sim Exp\left(\frac{1}{n\alpha}\right)\)
• Probability that actor \(i\) will be the next one to act is \(\frac{1}{n}\), where \(n\) is the number of actors in the network

Transition Steps

Objective Function

When actor \(i\) can change, it tries to maximize its objective function:

\[f_i(x, \mathbf{z}, \boldsymbol{\beta}) = \sum_k \beta_k s_{ik}(x, \mathbf{z})\]

• \(x\) is the network state
• \(\mathbf{z}\) is a matrix of actor-level covariates
• \(s_{ik}(x, \mathbf{z})\) are network & covariate statistics
• \(\beta_k\) are the parameters to be fit in the model
• There can be any number of \(\beta_k\) added to the model (problematic)

Transition Probability

The probability that actor \(i\) will change the tie to actor \(j\) is:

\[p_{ij} = \frac{\exp\left\{f_i(x(i\leadsto j), \mathbf{z}, \boldsymbol{\beta})\right\}}{\sum_h \exp\left\{f_i(x(i\leadsto h), \mathbf{z}, \boldsymbol{\beta})\right\}}\] - \(x(i\leadsto j)\) is the network identical to the current state, \(x\), except for \(x_{ij}\), which becomes \(1-x_{ij}\)

Transition Probability Heatmap

(Some) Possible Model Parameters

(Some) Possible Model Parameters

Network Effect	Interpretation	Visualization
outdegree*	Popularity
reciprocity*	Reciprocated relationships
transitive triplets	Your friend becomes my friend

Covariate Effect	Interpretation	Visualization
covariate-alter	Effect of my friend’s behavior on friendship
covariate-ego	Effect of my behavior on friendship
same covariate	Birds of a feather flock together

Small SAOMs Example

• Our small subset of the 50 friends data
• Fit three models for network evolution
• M1: “straw man” model that only has 2 parameters in objective function

All Microsteps

Three simulations from wave 1 to wave 2 according to M1: edges on y axis ordered by number of occurences in simulation

Summary Network

Simulate Wave 2 from Model 1 1000 times. Show edges that show up more than 5% of the time.

Other models

Chose additional parameters because of very low \(p\)-values (\(p\) < 0.001) according to RSiena (Ripley et al 2016)

• M2: M1 with one significant covariate effect added (drinking behavior encourages transitive ties)
• M3: M1 with one significant structural effect added (discourages transitive ties, encourages social distance)

Some Questions:

1. How do additional parameters affect the model fit?
2. Can we see the significance of these parameters?

Distributions of Fitted Model Effects

Correlation of Estimates

Other Summary Networks

Conclusion

• Can use visualization to learn about the model
• “Model Visualization” (Wickham et al 2015)
  • Model in the data space and data in the model space
  • Collections are more informative than singletons
  • Explore the process, not just the end result